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LOCALLY CONSERVATIVE AND FLUX CONSISTENT ITERATIVE METHODS

Linders, Viktor LU and Birken, Philipp LU (2024) In SIAM Journal on Scientific Computing 46(2). p.424-444
Abstract

Conservation and consistency are fundamental properties of discretizations of conservation laws, necessary to ensure physically meaningful solutions. In the context of systems of nonlinear hyperbolic conservation laws, conservation and consistency additionally play an important role in convergence theory via the Lax-Wendroff theorem. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are used to prove an extension of the Lax-Wendroff theorem incorporating pseudotime iterations with explicit Runge-Kutta methods. This result reveals that lack of flux consistency implies convergence towards weak solutions of a time dilated system of... (More)

Conservation and consistency are fundamental properties of discretizations of conservation laws, necessary to ensure physically meaningful solutions. In the context of systems of nonlinear hyperbolic conservation laws, conservation and consistency additionally play an important role in convergence theory via the Lax-Wendroff theorem. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are used to prove an extension of the Lax-Wendroff theorem incorporating pseudotime iterations with explicit Runge-Kutta methods. This result reveals that lack of flux consistency implies convergence towards weak solutions of a time dilated system of conservation laws, where each equation is modified by a particular scalar factor multiplying the spatial flux terms. Local conservation is further established for Krylov subspace methods with and without restarts, and for Newton's method under certain assumptions on the discretization. It is thus shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2 dimensional compressible Euler equations corroborate the theoretical results. A simple technique for enforcing flux consistency of Newton-Krylov methods is presented. Experiments indicate that its efficacy is case dependent, and diminishes as the number of iterations grow.

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type
Contribution to journal
publication status
published
subject
keywords
conservation laws, iterative methods, Lax-Wendroff theorem, Newton-Krylov methods, pseudotime iterations
in
SIAM Journal on Scientific Computing
volume
46
issue
2
pages
424 - 444
publisher
Society for Industrial and Applied Mathematics
external identifiers
  • scopus:85192684276
ISSN
1064-8275
DOI
10.1137/22M1503348
language
English
LU publication?
yes
id
4685559c-affc-4a6b-88ee-984d864e920b
date added to LUP
2024-05-28 15:47:01
date last changed
2024-05-28 15:47:12
@article{4685559c-affc-4a6b-88ee-984d864e920b,
  abstract     = {{<p>Conservation and consistency are fundamental properties of discretizations of conservation laws, necessary to ensure physically meaningful solutions. In the context of systems of nonlinear hyperbolic conservation laws, conservation and consistency additionally play an important role in convergence theory via the Lax-Wendroff theorem. Here, these concepts are extended to the realm of iterative methods by formally defining locally conservative and flux consistent iterations. These concepts are used to prove an extension of the Lax-Wendroff theorem incorporating pseudotime iterations with explicit Runge-Kutta methods. This result reveals that lack of flux consistency implies convergence towards weak solutions of a time dilated system of conservation laws, where each equation is modified by a particular scalar factor multiplying the spatial flux terms. Local conservation is further established for Krylov subspace methods with and without restarts, and for Newton's method under certain assumptions on the discretization. It is thus shown that Newton-Krylov methods are locally conservative, although not necessarily flux consistent. Numerical experiments with the 2 dimensional compressible Euler equations corroborate the theoretical results. A simple technique for enforcing flux consistency of Newton-Krylov methods is presented. Experiments indicate that its efficacy is case dependent, and diminishes as the number of iterations grow.</p>}},
  author       = {{Linders, Viktor and Birken, Philipp}},
  issn         = {{1064-8275}},
  keywords     = {{conservation laws; iterative methods; Lax-Wendroff theorem; Newton-Krylov methods; pseudotime iterations}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{424--444}},
  publisher    = {{Society for Industrial and Applied Mathematics}},
  series       = {{SIAM Journal on Scientific Computing}},
  title        = {{LOCALLY CONSERVATIVE AND FLUX CONSISTENT ITERATIVE METHODS}},
  url          = {{http://dx.doi.org/10.1137/22M1503348}},
  doi          = {{10.1137/22M1503348}},
  volume       = {{46}},
  year         = {{2024}},
}